Rational expressions are algebraic expressions that can be written as the ratio of two polynomials. Much like fractions in basic arithmetic, these expressions can often be simplified to present them in a more digestible form. Simplifying these expressions is crucial in maths, especially when solving equations or evaluating functions. This process involves factorising both the numerator and the denominator and then reducing the expression to its simplest form.
Factorising Numerator and Denominator
Factorising is the process of breaking down an expression into its simplest multipliers. When dealing with rational expressions, it's essential to factorise both the numerator and the denominator.
Techniques for Factorising
- Common Factor: This is the simplest method. If both terms have a common factor, factor it out.
- Example: For the expression 6x2 divided by 12x, the common factor is 6x. When factored, it becomes x divided by 2.
- Difference of Squares: If the expression is in the form a2 minus b2, it can be factored as (a + b) times (a - b).
- Example: x2 minus 9 can be factored as (x + 3) times (x - 3).
- Trinomial Factorisation: This involves breaking down a trinomial into two binomial expressions.
- Example: x2 + 5x + 6 can be factored as (x + 2) times (x + 3).
Simplifying the Expression
Once you've factorised both the numerator and the denominator, look for common factors that can be cancelled out.
- Example: (x2 + 5x + 6) divided by (x2 + 3x) becomes (x + 2) times (x + 3) divided by x times (x + 3). The common factor (x + 3) can be cancelled out, leaving (x + 2) divided by x.
Simplifying Complex Fractions
Complex fractions are fractions where the numerator, the denominator, or both contain fractions. Simplifying complex fractions makes them easier to work with.
Steps to Simplify Complex Fractions
1. Find the Least Common Denominator (LCD): Identify the denominators of the smaller fractions and determine their LCD.
2. Multiply Every Term by the LCD: This will eliminate the smaller fractions.
3. Simplify: Once the smaller fractions are eliminated, simplify the resulting expression if possible.
Example: Consider the complex fraction (2 divided by 3) divided by (4 divided by 5). The LCD of 3 and 5 is 15. Multiply every term by 15. This gives 10 divided by 12, which simplifies to 5 divided by 6.
Importance of Simplifying Rational Expressions
Simplifying rational expressions is not just a mathematical exercise; it has practical applications. For instance:
- Solving Equations: When dealing with equations involving rational expressions, simplifying can make the equation easier to solve.
- Graphing: Simplified expressions are often easier to graph as they provide a clearer view of the function's behaviour.
- Understanding Mathematical Models: In real-world applications, rational expressions can model various phenomena. A simplified expression can offer a clearer understanding of the model.
Example Questions
1. Simplify the rational expression (3x^3 minus 27x) divided by (9x^2).
Solution: First, factorise both the numerator and the denominator. Numerator: 3x times (x2 minus 9) = 3x times (x + 3) times (x - 3) Denominator: 9x2 Now, divide the numerator by the denominator: (3x times (x + 3) times (x - 3)) divided by 9x2. On simplifying, we get (x + 3) times (x - 3) divided by 3x.
2. Simplify the complex fraction (2 divided by 3) divided by (4 divided by 5).
Solution: To simplify, find the LCD of 3 and 5, which is 15. Multiply every term by 15. This gives 10 divided by 12, which simplifies to 5 divided by 6.
FAQ
A complex fraction is a fraction where the numerator, the denominator, or both contain fractions. To simplify a complex fraction, you can use the method of multiplying the top and bottom by the least common denominator (LCD) of the fractions within the complex fraction. This method helps eliminate the fractions in the numerator and denominator, making it easier to simplify the overall expression. Another approach is to simplify the numerator and denominator separately and then simplify the resulting fraction.
The denominator of a rational expression cannot be zero because division by zero is undefined in mathematics. If the denominator of a rational expression is zero for some value of the variable, then the expression is not defined for that value. This is because dividing by zero would lead to results that are inconsistent and contradictory within the framework of arithmetic. In the context of rational functions, points where the denominator is zero, but the numerator isn't, are called "vertical asymptotes".
Yes, all polynomial expressions are also rational expressions. This is because a polynomial can be thought of as a rational expression where the denominator is 1. Since the denominator is a polynomial (in this case, a constant polynomial), and the numerator is also a polynomial, the entire expression qualifies as a rational expression. However, it's essential to note that while all polynomials are rational expressions, not all rational expressions are polynomials, especially if the denominator is not a constant.
A rational expression is a fraction in which both the numerator and the denominator are polynomials. For example, (x2 + 3x + 2) / (x2 - 1) is a rational expression. On the other hand, a rational function is a function which is defined by a rational expression. In other words, it's a function that assigns to each input value (from its domain) an output value based on a rational expression. The distinction is subtle: while every rational function corresponds to a rational expression, not every rational expression represents a function, especially if the denominator can be zero for some values of x.
To determine if a rational expression is undefined for certain values of x, you need to find the values of x that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for x. The solutions to this equation are the values of x for which the rational expression is undefined. It's crucial to check these values because they represent points where the function is not defined or has vertical asymptotes.
Practice Questions
To simplify the given rational expression, we'll first find a common denominator for the terms in the numerator. The term 7 can be expressed as 7y/y to have the same denominator as 1/y. This gives us:
Numerator: 7y + 1
The denominator remains x - y.
Thus, the simplified expression is: (7y + 1) / y(x - y).
To simplify this expression, we need to factorise both the numerator and the denominator.
The numerator x2 - 9 is a difference of squares and can be factorised as (x + 3)(x - 3).
The denominator x2 - 4x + 4 can be factorised as (x - 2)(x - 2) or (x - 2)2.
Thus, the simplified expression is: (x + 3)(x - 3) / (x - 2)2.
